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ECE201 Lect-41 Sinusoids (8.1); Phasors (8.3); Complex Numbers (Appendix) Dr. Holbert January 30, 2006

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ECE201 Lect-42 Introduction Any steady-state voltage or current in a linear circuit with a sinusoidal source is a sinusoid. –This is a consequence of the nature of particular solutions for sinusoidal forcing functions. –All steady-state voltages and currents have the same frequency as the source.

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ECE201 Lect-43 Introduction (cont.) In order to find a steady-state voltage or current, all we need to know is its magnitude and its phase relative to the source (we already know its frequency). Usually, an AC steady-state voltage or current is given by the particular solution to a differential equation.

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ECE201 Lect-44 The Good News! We do not have to find this differential equation from the circuit, nor do we have to solve it. Instead, we use the concepts of phasors and complex impedances. Phasors and complex impedances convert problems involving differential equations into simple circuit analysis problems.

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ECE201 Lect-45 Phasors A phasor is a complex number that represents the magnitude and phase of a sinusoidal voltage or current. Remember, for AC steady-state analysis, this is all we need---we already know the frequency of any voltage or current.

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ECE201 Lect-46 Complex Impedance Complex impedance describes the relationship between the voltage across an element (expressed as a phasor) and the current through the element (expressed as a phasor). Impedance is a complex number. Impedance depends on frequency.

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ECE201 Lect-47 Complex Impedance (cont.) Phasors and complex impedance allow us to use Ohms law with complex numbers to compute current from voltage, and voltage from current.

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ECE201 Lect-48 Sinusoids Period: T –Time necessary to go through one cycle Frequency: f = 1/T –Cycles per second (Hz) Angular frequency (rads/sec): = 2 f Amplitude: V M

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ECE201 Lect-49 Example What is the amplitude, period, frequency, and angular (radian) frequency of this sinusoid?

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ECE201 Lect-410 Phase

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ECE201 Lect-411 Leading and Lagging Phase x 1 (t) leads x 2 (t) by - x 2 (t) lags x 1 (t) by - On the preceding plot, which signals lead and which signals lag?

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ECE201 Lect-412 Class Examples Learning Extension E8.1 Learning Extension E8.2

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ECE201 Lect-413 Phasors A phasor is a complex number that represents the magnitude and phase of a sinusoidal voltage or current:

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ECE201 Lect-414 Phasors (cont.) Time Domain: Frequency Domain:

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ECE201 Lect-415 Summary of Phasors Phasor (frequency domain) is a complex number: X = z = x + jy Sinusoid is a time function: x(t) = z cos( t + )

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ECE201 Lect-416 Class Examples Learning Extension E8.3 Learning Extension E8.4

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ECE201 Lect-417 Complex Numbers x is the real part y is the imaginary part z is the magnitude is the phase z x y real axis imaginary axis

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ECE201 Lect-418 More Complex Numbers Polar Coordinates: A = z Rectangular Coordinates: A = x + jy

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ECE201 Lect-419 Are You a Technology Have? There is a good chance that your calculator will convert from rectangular to polar, and from polar to rectangular. Convert to polar: 3 + j4 and -3 - j4 Convert to rectangular: 2 45 & -2 45

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ECE201 Lect-420 Arithmetic With Complex Numbers To compute phasor voltages and currents, we need to be able to perform computation with complex numbers. –Addition –Subtraction –Multiplication –Division

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ECE201 Lect-421 Complex Number Addition and Subtraction Addition is most easily performed in rectangular coordinates: A = x + jyB = z + jw A + B = (x + z) + j(y + w) Subtraction is also most easily performed in rectangular coordinates: A - B = (x - z) + j(y - w)

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ECE201 Lect-422 Complex Number Multiplication and Division Multiplication is most easily performed in polar coordinates: A = A M B = B M A B = (A M B M ) ( ) Division is also most easily performed in polar coordinates: A / B = (A M / B M ) ( )

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ECE201 Lect-423 Examples Find the time domain representations of V = 104V - j60V I = -1mA - j3mA at 60 Hz If Z = -1 + j2 k, then find the value of I Z + V

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Phasors A phasor is a complex number that represents the magnitude and phase of a sinusoid:

Phasors A phasor is a complex number that represents the magnitude and phase of a sinusoid:

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